Metadata
bibkey: cfmp2026cycledegree9index72obstruction
authors: trureturing contributors
year: 2026
title: No normal torsionfree index-72 orientation quotient for the degree-(8,9) Coxeter sector
doi: null
url: https://github.com/the-omega-institute/trureturing/pull/12077
claim: No normal torsionfree Gamma normal in W+=[4,9,3] with [W+:Gamma]=72; therefore the regular six-sector Coxeter-cover route cannot realize N=24.
license: citation-only
triage: anchor
strata_touched: []
For the six-sector Coxeter block KQ with W+ the orientation subgroup and K=<c,d> of order six, N=[W+:Gamma]/3. Assuming Gamma is normal and torsionfree with index 72, the quotient has order 72. Restricting to the subgroup generated by r=bc (order 9), s=cd (order 3), and t=db=(rs)^(-1) (order 2), Sylow-3 action gives a contradiction in both possible cases |H|=36 and |H|=72; |H|=18 is excluded by its unique Sylow-3 subgroup. The order-72 case would give an order-24 quotient of Delta(3,3,2)=A4, impossible. The order-36 case gives a central z=r^3 and an A4 conjugacy-lift contradiction because r has order 9 while s has order 3.
This is only an obstruction to normal (regular-cover) Coxeter-sector quotients. It does not rule out non-normal covers or arbitrary pseudo-simplicial N=24 pairings.
Verified locator
- URL: https://github.com/the-omega-institute/trureturing/pull/12077